Yeah - this was my first hangout... not around here so much these days :-)
So you're just using an Access MDB application with linked tables?
Though the linked tables may seem to offer implied connectivity to the SQL database, they do so no more than the explicit connection to each individual table (based on that linked table's ODBC connection string).
Your CurrentProject.Connection object still points to the currently running instance of your Access FE application.
(Don't let the fact that it's an ADO object confuse the issue for you - ADO works perfectly well against Access/Jet too ;-)
Dim cnn As ADODB.Connection
Dim cmd As ADODB.Command
Dim rs As ADODB.Recordset
Set cnn = New ADODB.Connection
cnn.Open "Your Connection String"
Set cmd = New ADODB.Command
Set cmd.ActiveConnection = cnn
cmd.CommandText = "NewCaseInsertDocs"
cmd.CommandType = adCmdStoredProc
'Set the parameters
Set rs = New ADODB.Recordset
rs.Open cmd, , adOpenKeyset, adLockOptimistic
Where "Your Connection String" is obviously formed based on your own setup - similar even to your linked table connectionstring if you like. You'll have built them often enough in your ASP.NET apps no doubt.
Note the change to adOpenKeyset - as that's the cursor type you'd get back anyway with the given code. (Not static).
If all you wanted was a forward only, read only recordset - then the original code you had would be fine.
i.e. Set rs = cmd.Execute
Note also that with such an execution - there's no need to instantiate the recordset first.